Aircraft: CITATION EXCEL (C560XL)
Phase of flight: Initial Approach
Anomaly: Deviation - Altitude Excursion From Assigned Altitude; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance; Flight Deck / Cabin / Aircraft Event Other / Unknown; Inflight Event / Encounter Unstabilized Approach

This situation occurred While in IMC conditions; and being vectored onto an ILS approach into BJC. We had just checked on with DEN approach and were issued a descent. As I was descending; the PM began turning anti-ice on; and became distracted discussing anti-ice and why they felt it was necessary to have it on at a higher temperature (OAT was 11C and anti-ice was not required). This unnecessary explanation caused the flight crew to miss part of a radio call from ATC. I was able to catch part of our call sign; and the fact that a heading was being assigned; but I didn't catch the actual heading. After getting back the PMs attention to the radio calls; I informed them that the last radio call was for us; and to query the heading assigned. PM replied to ATC with 'please repeat.'ATC did not reply to PM's request; and instead told us we were '5 miles from ALIKE; maintain 7000 until established; cleared for the ILS 30R approach.' Since we did not have a proper heading yet; but were cleared for the approach; I rolled the heading bug to an intercept heading that I thought would capture the localizer and selected APPROACH mode; in an effort to move us into the proper direction. At this time; PM replied to ATC's radio call with; 'Maintain 5000 until established;' and rolled the ASEL to 5000; as we were still descending to 7000. ATC replied; 'negative - maintain 7000' - PM repeated to 'maintain 7000' I then realized the PM was lagging behind the procedure we were flying; so I rolled the ASEL back to 7000 as the PM read back the instructions to maintain 7000. It was at this point I should have disconnected autopilot and hand-flown the ILS approach we were cleared for; and as I heard it; instead of attempting to correct the PM's actions. After the PM finished their radio call; I immediately made the radio call to clarify the heading ATC originally assigned us; as the PM still had not retrieved that information. ATC replied with the heading; and then indicated we had flown through the localizer; so he assigned a new heading; and asked us to slow our airspeed. Shortly after that; ATC canceled the approach clearance; and gave us a left turn to heading 200; and climb & maintain 8000. At this point we were no longer in a position to make a stabilized approach. I disconnected the autopilot and we complied with this new instruction. We were given new vectors back onto the approach and landed safely; with happy passengers who; fortunately; had no idea any of this had occurred. There was a breakdown in communication between flight crew due to an unnecessary discussion about anti-ice usage at a time where a sterile cockpit is required. This breakdown of communication and loss of proper monitoring with ATC caused the need to be re-vectored back to the ILS. The missed vector could have been considered a Pilot Deviation by ATC though nothing was said to us about it. The approach became unstable due to a breakdown in Crew CRM; which led to missed calls and misunderstandings with ATC. I should have become aware that PM was becoming overwhelmed with the approach; and taken over flying the aircraft earlier in the approach; so PM's inputs would not have affected the path of the aircraft. In a post-flight debrief we discussed our communication breakdown & determined a better job could have been done in not discussing issues not immediately pertinent to the phase of flight. This is especially important during the high workload we were experiencing at the time (IMC; vectors to an approach; descending; slowing & configuring). As the PIC I should have been more forceful with ending the anti-ice discussion so the PM could focus on the tasks at hand. I should have more quickly initiated the radio call to ATC to confirm the missed vector and assigned altitude when it was clear the PM was not situationally aware of what needed to be done. I will be sure to take all these lessons learned and apply them to my future flights. CE560XL Captain reported executing a go-around when the approach became unstabilized following confusion in the cockpit as to the ATC clearance.